Changes in the adiabatic invariant and streamline chaos in confined incompressible Stokes flow

The steady incompressible flow in a unit sphere introduced by Bajer and Moffatt [J. Fluid Mech. 212, 337 (1990)] is discussed. The velocity field of this flow differs by a small perturbation from an integrable field whose streamlines are almost all closed. The unperturbed flow has two stationary saddle points (poles of the sphere) and a two-dimensional separatrix passing through them. The entire interior of the unit sphere becomes the domain of streamline chaos for an arbitrarily small perturbation. This phenomenon is explained by the nonconservation of a certain adiabatic invariant that undergoes a jump when a streamline crosses a small neighborhood of the separatrix of the unperturbed flow. An asymptotic formula is obtained for the jump in the adiabatic invariant. The accumulation of such jumps in the course of repeated crossings of the separatrix results in the complete breaking of adiabatic invariance and streamline chaos. (c) 1996 American Institute of Physics.     

Adiabatic chaos in a two-dimensional mapping

A close to identity symplectic mapping describing the dynamics of a charged particle in the field of an infinitely wide packet of electrostatic waves is studied. A region of chaotic dynamics, whose width is large for an arbitrarily small deviation of the mapping from the identity, exists on the phase cylinder. This is explained by the quasirandom change occurring in an adiabatic invariant of the problem when the phase trajectory crosses a resonance curve. An asymptotic formula is derived for the jump in the adiabatic invariant. The width of the chaos region and the density of the set of invariant curves near the boundary of the chaos region are estimated. (c) 1996 American Institute of Physics.     

Geometry of a chaotic layer

An analysis is made of the dependence of the geometric shape of the chaotic layer near the separatrix of a nonlinear resonance of a Hamiltonian system on the parameters of this system. A separatrix algorithmic mapping, which describes the motion near the separatrix in the presence of an asymmetric perturbation having an arbitrary degree of asymmetry. The separatrix algorithmic mapping is an algorithm containing conditional transfer instructions, is considered. An analytic procedure is derived to reduce the separatrix algorithmic mapping to the unified surface of the cross section of the initial Hamiltonian system (mapping synchronization procedure). It is observed that in the case of the high-frequency perturbation λ → +∞ (where λ is the ratio of the perturbation frequency to the frequency of small phase oscillations at resonance), the chaotic layer is subjected to strong bending in the sense that during motion near the separatrix theamplitude of the energy deviations relative to the unperturbed separatrix value is much larger than the layer width. However, the synchronized separatrix algorithmic mapping ensures an accurate representation of the phase portrait of the layer for both low and high values of the parameter λ provided that the amplitude of the perturbation is fairly small. This is demonstrated by comparing the phase portraits obtained using the synchronized separatrix algorithmic mapping with the results of direct numerical integrations of the initial Hamiltonian system.     

Accuracy of the conservation of the adiabatic invariant of a charged particle in plane waves

The conservation of the adiabatic invariant of a charged particle, entrapped by a transverse plane electromagnetic wave, is studied using a well-known method, both for the case when the phase velocity of the wave does not exceed the velocity of light, and for the opposite case. An expression is obtained for the variation of the adiabatic invariant, from which it is seen that the latter is conserved with exponential accuracy in the region of its existence.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 61–66, April, 1979.     

Probability phenomena due to separatrix crossing

The effect of a small or slow perturbation on a Hamiltonian system with one degree of freedom is considered. It is assumed that the phase portrait ("phase plane") of the unperturbed system is divided by separatrices into several regions and that under the action of the perturbations phase points can cross these separatrices. The probabilistic phenomena are described that arise due to these separatrix crossings, including the scattering of trajectories, random jumps in the values of adiabatic invariants, and adiabatic chaos. These phenomena occur both in idealized problems in classical mechanics and in real physical systems in planetary science and plasma physics contexts.     

On the absence of stable periodic orbits in domains of separatrix crossings in nonsymmetric slow-fast Hamiltonian systems

We consider a two degree of freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. We assume that at frozen values of the slow variables there is a separatrix on the phase plane of the fast variables and there is a region in the phase space (the domain of separatrix crossings) where projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under rather general conditions, we prove that there are no stable periodic trajectories of any prescribed period inside the domain of separatrix crossings, except maybe for periodic trajectories passing anomalously close to the saddle point.     

Suppression of dynamic chaos in Hamiltonian systems

The paper describes the results of a recent numerical study on the canonical mapping with a sawtooth force. The dynamic effects of the formation of invariant resonance structures of various orders, whose presence prevents the development of global chaos and restricts momentum diffusion in the phase space, are discussed. The dynamic situation near an integer resonance separatrix in the neighborhood of the critical state is studied, and the conditions responsible for the stability of this separatrix in the critical state are determined. Along with the mapping, the related continuous Hamiltonian system is considered. For this system, the separatrix mapping and the Mel’nikov-Arnold integral are introduced, whose analysis facilitates understanding the reasons responsible for the unusual dynamics. This dynamics is shown to be preserved under substantial saw shape changes. Relevant new problems and open questions are formulated.     

Adiabatic theorem for finite dimensional quantum mechanical systems

A new simple proof of the adiabatic theorem is given in the finite dimensional case for nondegenerate as well as degenerate states. An explicitly integrable two-level system is considered as an example. It is demonstrated that the error estimate given by the adiabatic theorem cannot be improved.     

Contribution to the symplectic structure in the quantization rule due to noncommutativity of adiabatic parameters

A geometric construction of the `ala Planck action integral (quantization rule) determining adiabatic terms for fast-slow systems is considered. We demonstrate that in the first (after zero) adiabatic approximation order, this geometric rule is represented by a deformed fast symplectic 2-form. The deformation is controlled by the noncommutativity of the slow adiabatic parameters. In the case of one fast degree of freedom, the deformed symplectic form incorporates the contraction of the slow Poisson tensor with the adiabatic curvature.The same deformed fast symplectic structure is used to represent the improved adiabatic invariant in a geometric form.     

Quantum adiabatic theorem and universal holonomy

We give a simplified proof of the quantum adiabatic theorem for a system of possibly degenerate Hamiltonians by taking Berry's phase into account. We also relate the adiabatic transformation to the parallel transport induced by the holonomy in the universal bundle over a Grassman manifold. The special case of a nondegenerate Hamiltonian is precisely the cyclic quantum evolution studied by Aharanov and AnandanThe author is S. Y. Wu     

相空间中非完整可控力学系统的对称性摄动与绝热不变量

研究相空间中非完整可控力学系统的对称性摄动与绝热不变量. 列出相空间中未受扰非完整可控力学系统的形式不变性导致的Noether守恒量. 基于力学系统高阶绝热不变量的定义，研究小扰动作用下相空间中非完整可控力学系统的形式不变性摄动与绝热不变量，给出了精确不变量与绝热不变量存在的条件与形式，并举例说明结果的应用.     

A new type of adiabatic invariants for nonconservative systems of generalized classical mechanics

The perturbations to symmetries and adiabatic invariants for nonconservative systems of generalized classical mechanics are studied. The exact invariant in the form of Hojman from a particular Lie symmetry for an undisturbed system of generalized mechanics is given. Based on the concept of high-order adiabatic invariant in generalized mechanics, the perturbation to Lie symmetry for the system under the action of small disturbance is investigated, and a new adiabatic invariant for the nonconservative system of generalized classical mechanics is obtained, which can be called the Hojman adiabatic invariant. An example is also given to illustrate the application of the results.     

El-Nabulsi动力学模型下Birkhoff系统Noether对称性的摄动与绝热不变量

基于El-Nabulsi动力学模型,研究了小扰动作用下Birkhoff系统Noether对称性的摄动与绝热不变量问题.首先,将El-Nabulsi提出的在分数阶微积分框架下基于Riemann-Liouville分数阶积分的非保守系统动力学模型拓展到Birkhoff系统,建立El-Nabulsi-Birkhoff方程;其次,基于在无限小变换下El-Nabulsi-Pfaff作用量的不变性,给出Noether准对称性的定义和判据,得到了Noether对称性导致的精确不变量;再次,引入力学系统的绝热不变量概念,研究El-Nabulsi动力学模型下受小扰动作用的Birkhoff系统Noether对称性的摄动与绝热不变量之间的关系,得到了对称性摄动导致的绝热不变量的条件及其形式.作为特例,给出了El-Nabulsi动力学模型下相空间中非保守系统和经典Birkhoff系统的Noether对称性的摄动与绝热不变量.以著名的Hojman-Urrutia问题为例,研究其在El-Nabulsi动力学模型下的Noether对称性,得到了相应的精确不变量和绝热不变量.     

Transition to chaos in the conservative four-wave parametric interactions

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.     

非保守动力学系统Noether对称性的摄动与绝热不变量

基于非保守系统的El-Nabulsi动力学模型, 研究了非保守动力学系统Noether对称性的摄动与绝热不变量问题.首先, 引入El-Nabulsi在分数阶微积分框架下基于Riemann-Liouville分数阶积分提出的类分数阶变分问题, 列出非保守系统的Euler-Lagrange方程; 其次, 给出了Noether准对称变换的定义和判据, 建立了Noether对称性与不变量之间的关系, 得到了精确不变量; 最后, 提出并研究了该系统受小扰动作用后Noether对称性的摄动与绝热不变量问题, 证明了绝热不变量存在的条件及形式, 并举例证明结果的应用. 关键词： 非保守系统 El-Nabulsi动力学模型 对称性摄动 绝热不变量     

Passive scalars,three-dimensional volume-preserving maps,and chaos

The dynamics of a medium-sized particle (passive scalar) suspended in a general time-periodic incompressible fluid flow can be described by three-dimensional volume-preserving maps. In this paper, these maps are studied in limiting cases in which some of the variables change very little in each iteration and others change quite a lot. The former are called slow variables or actions, the latter fast variables or angles. The maps are classified by their number of actions. For maps with only one action we find strong evidence for the existence of invariant surfaces that survive the nonlinear perturbation in a KAM-like way. On the other hand, for the two-action case the motion is confined to invariant lines that break for arbitrary small size of the nonlinearity. Instead, we find that adiabatic invariant surfaces emerge and typically intersect the resonance sheet of the fast motion. At these intersections surfaces are locally broken and transitions from one to another can occur. We call this process, which is analogous to Arnold diffusion, singularity-induced diffusion. It is characteristic of two-action maps. In one-action maps, this diffusion is blocked by KAM-like surfaces.On leave of absence from the Departamento de Fisica, Universidad Nacional de La Plata, (1990), La Plata, Argentina.     

On the motion of non-linear oscillators with a fractional-order restoring force and time variable parameters

An analytical approach to determine the approximate solution for the periodic motion of non-conservative oscillators with a fractional-order restoring force and slowly varying parameters is presented. The solution has the form of the first-order differential equation for the amplitude and phase of motion. The method used is based on the combination of the Krylov-Bogoliubov method with Hamilton's variational principle with the uncommutative rule for the variation of velocity. The conservative systems with slowly varying parameters are also considered. The corresponding adiabatic invariant is obtained. Two examples are given to illustrate derived theoretical results.     

Perturbation to Noether-Mei Symmetry and Adiabatic Invariants for Nonholonomic Mechanical Systems in Phase Space

Based on the concept of adiabatic invariant, the perturbation to Noether Mei symmetry and adiabatic invariants for nonholonomie mechanical systems in phase space are studied. The definition of the perturbation to Noether-Mei symmetry for the system is presented, and the criterion of the perturbation to Noether-Mei symmetry is given. Meanwhile, the Noether adiabatic invariants and the Mei adiabatic invariants for the perturbed system are obtained.